1. Field of the Invention
This invention is an electronic apparatus for damping roll/yaw nutation within the control deadband of a satellite which is three-axis stabilized to local vertical by means of a pitch-biased, body-fixed reaction wheel and offset roll/yaw thrusters.
2. Description of Prior Art
The present invention can perhaps be best understood be reference to a typical prior art apparatus, the WHECON (WHEel CONtrol) system, described in AIAA Paper No. 68-461, "Analysis and Design of WHECON -- An Attitude Control Concept" by H. J. Doherty, E. D. Scott and J. J. Rodden, given at the AIAA Second Communications Satellite Systems Conference, San Francisco, California Apr. 8-10, 1968.
Briefly, the WHECON system operates as follows:
Referring to FIG. 1 of the present drawing, we define three mutually-orthogonal "local-tangent" reference axes. This reference frame is not satellite-fixed, i.e., the orientation of any line in the satellite with respect to this reference frame changes as the instantaneous attitude of the satellite changes with respect to nominal attitude. The x-axis is parallel to the spacecraft velocity vector (again, regardless of the spacecraft orientation); the y-axis is normal to the satellite orbit plane; and the z-axis is radial to the orbit plane, i.e., it is directed toward the center of the earth.
We also define a "satellite-fixed" reference frame whose axes are aligned with the principal axes of the satellite (originating at its center of mass) and are fixed with respect to the satellite regardless of its instantaneous orientation. This satellite fixed frame is aligned with the local tangent frame, by definition, whenever the satellite attitude is nominal, i.e., where there is not roll, pitch or yaw. Under conditions of attitude error, the two reference frames will be misaligned.
The reaction wheel is mounted to the satellite so that its angular momentum vector lies along the negative fixed y-axis (pitch axis).
A condition of satellite yaw consists of angular displacement of the satellite-fixed x- and y-axes, respectively, from the local tangent x- and y-axes. Likewise, roll consists of angular displacement of the satellite-fixed y- and z-axes, respectively, from the local tangent y- and z-axes. Finally, pitch consists of angular displacement of the satellite-fixed x- and z-axes, respectively, from the local tangent x- and z-axes.
Satellite pitch motion decouples dynamically from roll/ yaw motion, and current art in reaction wheel pitch attitude control provides adequate means for controlling it. Accordingly, all subsequent discussion will relate to roll/yaw attitude motion and its control.
What constitutes satellite roll at a particular point in the earth orbit will constitute yaw 1/4 revolution about the earth later. Likewise, what constitutes satellite yaw at a particular point in the earth orbit will constitute roll 1/4 revolution later. These conditions will interchange every quarter revolution. It is clear, therefore, that a roll sensor (e.g., an ordinary horizon sensor) can be used as a basis for detection and control of both roll and yaw. Satellite roll/yaw attitude motion consists of two quasi-periodic components -- one (precession) at a low frequency of approximately orbit rate and the other (nutation) at a much higher frequency, related to the satellite inertia properties and angular momentum of the reaction wheel. In the absence of any attitude control, the total satellite angular momentum vector will rotate in a "coning" fashion at nutation frequency about an apex axis which moves, with respect to the local tangent y-axis, with a combination of constant, secular (i.e., linear in time), and orbit rate motion. The magnitudes of the various components depend on the magnitudes of external torques, such as solar radiation, atmospheric drag, and meteorite collisions. Since this attitude motion represents misalignment of the satellite-fixed frame with respect to the local tangent reference frame, some means must be provided for controlling the motion to force the satellite frame to realign with the local tangent frame.
In the WHECON system this is accomplished by means of an opposed pair of thrusters whose line of thrust is parallel to the satellite-fixed y-axis and, as shown in FIG. 1, offset from the z-axis by an angle .alpha.. Because of this offset, activation of either of the thrusters will create simultaneous roll and yaw torques on the satellite. This, together with the natural gyroscopic coupling of roll and yaw, makes it possible to control both roll and yaw with a single roll error actuated pulse modulating controller.
A "derived rate" modulation controller circuit which may be used in connection with the WHECON system is shown in FIG. 2. Here the error signal from the horizon sensor 18 is passed through a low-pass filter 20 to remove high frequency sensor noise. The filtered signal is passed through a summing junction 25 where it is broken into two branches.
One branch is fed into a positive 2-state switch with hysteresis, i.e., a "Schmitt trigger", 30P, which is activated if the filtered error signal is sufficiently positive, indicating out-of-deadband roll/yaw attitude; the other is fed into a negative Schmitt trigger 30N which is activated if the signal is sufficiently negative. The positive Schmitt trigger, while open, activates a one-shot multivibrator 35P, and the output of the multivibrator is gated together with the Schmitt trigger output to operate the negative thruster valve 40N. The same final signal is passed through another summing junction 45 to the integrator 50, whose output is fed into the first summing junction 25. When the integrator has accumulated a large enough value to indicate that the negative thruster valve has been operating a sufficient time to correct the roll error by returning the roll/yaw attitude to the deadband, the output of the integrator will counterbalance the input from the horizon (roll attitude error) sensor and deactivate the positive Schmitt trigger 30P which will, in turn, deactivate the negative thruster 40N.
The lower loop of the circuit shown in FIG. 2, which is utilized to correct a negative roll attitude error by operating the positive thruster valve 40P, is entirely similar in operation to the upper loop.
The expressions shown within the filter 20 and the integrator 50, represent, respectively, Laplace transforms of the filter transfer function and feedback integrator. The values of the parameters to be used depend on the vehicle characteristics and performance requirements, and are readily determined by one skilled in the art, for example, by following the guidelines set forth in Section IV (Design Methodology), "Precision Attitude Control with a Single Body-Fixed Momentum Wheel", R. P. Iwens, A. W. Fleming, V. A. Spector, Paper No. 74-894, AIAA Mechanics and Control of Flight Conference, Aug. 1974.
The object of the control circuit is to operate the proper thruster so that its operation, in conjunction with the gyroscopic nature of the reaction wheel 10 and the dynamics of the satellite as it orbits the earth, will eliminate roll and/or yaw errors by causing the orientation of the satellite to return within the roll attitude deadband. The deadband is an angle, calculated according to the orientation accuracy requirements of the particular satellite, representing the range from the nominal within which the roll/yaw attitude of the satellite may vary without the necessity of corrective action.
The WHECON system works well for stabilizing large excursions in roll error outside the controller deadband. However, for roll motion very near or within the deadband the controller will provide only very short minimum impulse pulses (determined by the time constant of the one-shot multivibrators 35P and 35N). The effective control damping produced by the feedback integrator 50 is then negligible. With the WHECON system and thrusters sized to produce satisfactory response for large attitude motions, undamped motion within the deadband will tend to develop into a hard two sided nutational limit cycle across the deadband in which the motion will "strike" one "wall" of the deadband, causing a thruster to be activated to reverse the motion; then strike the opposite wall causing the thruster of opposite sense to be activated; then return to the first wall, and so on in an oscillatory manner. This limit cycle causes many unnecessary thruster actuations and is undesirable for propellant economy and thruster reliability.